On the Unique Games Conjecture

1
On the Unique Games Conjecture

• Subhash Khot
• Georgia Inst. Of Technology.
• At FOCS 2005

2
NP-hard Problems

• Vertex Cover
• MAX-3SAT
• Bin-Packing
• Set Cover
• Clique
• MAX-CUT
• ..
• ..

3
Approximability Algorithms

• A C-approximation algorithm computes (C gt 1),
• for problem instance I , solution A(I) s.t.
• Minimization problems
• A(I) ? C ? OPT(I)
• Maximization problems
• A(I) ? OPT(I) / C

4
Some Known Approximation Algorithms

• Vertex Cover 2 - approx.
• MAX-3SAT 8/7 - approx. Random
  assignment.
• Packing/Scheduling (1?) approx. ? ? gt 0
  (PTAS)
• Set Cover ln n approx.
• Clique n/log n Boppana
  Halldorsson92
• Many more , ref. Vazirani01

5
PCP Theorem

• B85, GMR89, BFL91, LFKN92, S92,
  PY91
• FGLSS91, AS92 ALMSS92
• Theorem It is NP-hard to tell whether
  a MAX-3SAT
• instance is
• satisfiable (i.e. OPT
  1) or
• no assignment satisfies
  more than 99 clauses
• (i.e.
  OPT ? 0.99).
• i.e. MAX-3SAT is 1/0.99 1.01 hard to
  approximate.
• i.e. MAX-3SAT and MAX-SNP-complete problems
  PY91
• have no PTAS.

6
Approximability Towards Tight Hardness Results

• Hastad96 Clique n1-?
• Hastad97 MAX-3SAT 8/7 - ?
• Feige98 Set Cover (1- ?) ln n

Dinur05 Combinatorial Proof of PCP Theorem !
7
Open Problems in Approximability

• Vertex Cover
• (1.36 vs. 2) DinurSafra02
• Coloring 3-colorable graphs
• (5 vs. n3/14)
• KhannaLinialSafra93, BlumKarger97
• Sparsest Cut
• (1 vs. (logn)1/2) AroraRaoVazirani04
• Max Cut
• (17/16 vs 1/0.878 )
• Håstad97, GoemansWilliamson94
• ..

8
Unique Games Conjecture Khot02

• Implies these hardness results
• Vertex Cover 2- ?
  KR03
• Coloring 3-colorable ?(1)
  DMR05
• graphs (variant of UGC)
• MAX-CUT 1/0.878.. - ?
  KKMO04
• Sparsest Cut,
• Multi-cut
  KV05,
• 
  ?(1) CKKRS04
• Min-2SAT-Deletion
  K02, CKKRS04

9
Unique Games Conjecture

• Led to
• MOO05 Majority Is Stablest Theorem
• KV05 Negative type metrics do
  not embed
• into L1 with O(1)
  distortion.
• Optimal
  integrality gap for MAX-CUT
• SDP with Triangle
  Inequality.

10
Integrality Gap Definition

• Given Maximization Problem
• Specific SDP relaxation.
• For every problem instance G,
• SDP(G) ? OPT(G)
• Integrality Gap Max G SDP(G) / OPT(G)
• Constructing gap instance negative result.

11
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

12
Unique Games Conjecture

• A maximization problem called Unique Game is
  hard to approximate.
• Gap-preserving reductions from
• Unique Game ?
• Hardness results for Vertex Cover, MAX-CUT,
  Graph-Coloring, ..

13
Example of Unique Game

• OPT max fraction of equations that
  can
• be satisfied by any
  assignment.
• x1 x3 2
  (mod k)
• 3 x5 - x2 -1
  (mod k)
• x2 5 x1 0
  (mod k)

UGC ? For large k, it is NP-hard to
tell whether OPT ? 99
or OPT ?
1
14
2-Prover-1-Round Game (Constraint Satisfaction
Problem )
variables
constraints ?
15
2-Prover-1-Round Game (Constraint Satisfaction
Problem )
variables
k labels Here k4
constraints ?
16
2-Prover-1-Round Game (Constraint Satisfaction
Problem )
variables
k labels Here k4
Constraints Bipartite graphs or Relations ?
? k ? k
17
2-Prover-1-Round Game (Constraint Satisfaction
Problem )
Find a labeling that satisfies max
constraints
variables
k labels Here k4
OPT(G) 7/7
18
Hardness of Finding OPT(G)

• Given a 2P1R game G, how hard
• is it to find OPT(G) ?
• PCP Theorem Razs Parallel Repetition Theorem
  
• For every ?, there is integer k(?),
  s.t.
• it is NP-hard to tell whether a 2P1R
• game with k k(?) labels has
• OPT 1 or OPT ? ?

In fact k 1/poly(?)
19
Reductions from 2P1R Game

• Almost all known hardness results
• (e.g. Clique, MAX-3SAT, Set Cover, SVP,
  . )
• are reductions from 2P1R games.
• Many special cases of 2P1R games are known
  to be hard, e.g. Multipartite graphs,
• Expander graphs,
• Smoothness property, .

What about unique games ?
20
Unique Game 2P1R Game
with Permutations
variable
k labels Here k4
21
Unique Game 2P1R Game
with Permutations
variable
k labels Here k4
Permutations or matchings ? k ? k
22
Unique Game 2P1R Game
with Permutations
Find a labeling that satisfies max
constraints
OPT(G) 6/7
23
Unique Games

• Considered before
• Feige Lovasz92 Parallel Repetition of
  UG
• reduces OPT(G).
• How hard is approximating OPT(G)
• for a unique game G ?
• Observation Easy to decide whether
• OPT(G) 1.

24
MAX-CUT is Special Case of Unique Game

• Vertices Binary variables x, y, z, w, .
• Edges Equations x y 1 (mod 2)
• Hastad97
• NP-hard to tell whether
• OPT(MAX-CUT) ? 17/21
• or OPT(MAX-CUT) ? 16/21

25
Unique Games Conjecture

• For any ?, ?, there is integer k(?, ?), s.t.
• it is NP-hard to tell whether a Unique
• Game with k k(?, ?) labels has
• OPT ? 1- ?
• or OPT ? ?
• i.e. Gap-Unique Game (1- ? , ?)
• is NP-hard.

26
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

27
Case Study MAX-CUT

• Given a graph, find a cut that maximizes
• fraction of edges cut.
• Random cut 2-approximation.
• GW94 SDP-relaxation and rounding.
• min 0 lt ? lt 1 ? / (arccos
  (1-2?) / ? )
• 1/0.878 approximation.
  
• KKMO04 Assuming UGC, MAX-CUT is
• 1/0.878 - ? hard to
  approximate.

28
Reduction to MAX-CUT

• Unique Game Graph H
• Completeness
• OPT(UG) gt 1-o(1) ? ? ? - o(1)
  cut.
• Soundness
• OPT(UG) lt o(1) ? No cut
  with
• size arccos
  (1-2?) / ? o(1)
• Hardness factor ? / (arccos (1-2?) / ? ) -
  o(1)
• Choose best ? to get 1/0.878 (
  GW94)

29
Reduction from Unique Game

Gadget constructed via Fourier theorem
Connecting gadgets
via Unique Game instance DMR05 UGC
reduces the analysis of the
entire construction to the analysis
of the gadget.
Gadget Basic gadget ---gt Bipartite
gadget ---gt Bipartite gadget with permutation
30
Basic Gadget

• A graph on 0,1 k with specific
  properties
• (e.g. cuts, vertex covers, colorability)

x 011
k labels
0,1 k
Y 110
31
Basic Gadget MAX-CUT

• Weighted graph, total edge weight 1.
  
• Picking random edge
• x ?R 0,1 k
• y lt-- flip every co-ordinate
  of x with
• probability ? (? ?
  0.8)

x
32
MAX-CUT Gadget Co-ordinate Cut Along
Dimension i
Fraction of edges cut Pr(x,y) xi ? yi

? Observation These are the maximum cuts.
33
Bipartite Gadget

• A graph on 0,1 k ? 0,1 k (double cover
  of basic
• 
  gadget)

x 011
y 110
34
Cuts in Bipartite Gadget
0,1 k
0,1 k
Matching co-ordinate cuts have size ?
35
Bipartite Gadget with Permutation ? k -gt
k

• Co-ordinates in second hypercube permuted via
  ?.

Example ? reversal of co-ordinates.
? (y) 011
36
Reduction from Unique Game
37
Instance H of MAX-CUT
Bipartite Gadget via ?
38
Proving Completeness

• Unique Game Graph H

(Completeness) OPT(UG) gt 1-o(1)
? H has ? - o(1) cut.
39
Completeness OPT(UG) ? 1-o(1)
label 1
Labels 1,2,3
label 2
label 3
label 2
label 1
label 1
label 3
40

Completeness OPT(UG) ? 1-o(1)
0,1 k
Vertices
Edges
Hypercubes are cut along dimensions
labels. MAX-CUT ?? ? - o(1)
?
41
Proving Soundness

• Unique Game Graph H

(Soundness) OPT(UG) lt o(1)
? H has no cut of
size arccos (1-2?) / ?
o(1)
42
MAX-CUT Gadget

• Cuts Boolean functions f 0,1 k ?
  0,1
• Compare boolean functions
• that depend only on single co-ordinate
  vs
• where every co-ordinate has negligible
• influence (i.e. non-junta functions)

f(x1 x2 .. xk) xi
Influence (i, f) Prx f(x) ? f(xei)
f(x1 x2 .. xk) MAJORITY
43
Gadget Non-junta Cuts

• How large can non-junta cuts be ?
• i.e. cuts with all influences
  negligible ?
• Random Cut ½
• Majority Cut arccos (1-2?) / ?
  gt ½
• MOO05 Majority Is Stablest (Best)
• Any cut slightly
  better than
• Majority Cut must
  have
• influential
  co-ordinate.

44
Non-junta Cuts in Bipartite Gadget
0,1 k
0,1 k
MOO05 Any special cut with value
arccos (1-2?) / ? ? must
define a matching pair
of influential co-ordinates.
45
Non-junta Cuts in Bipartite Gadget
0,1 k
0,1 k
f 0,1 k --gt 0, 1
g 0,1 k --gt 0, 1
cut gt arccos (1-2?) / ? ? ?
? i Infl (i, f), Infl (i, g) gt ?(1)
46
Instance H of MAX-CUT
Bipartite Gadget via ?
47
Proving Soundness

• Assume arccos (1-2?) / ? ? cut exists.
• On ?/2 fraction of constraints, the
• bipartite gadget has arccos (1-2?) / ?
  ?/2 cut.
• ? matching pair of labels on this
  constraint.
• This is impossible since OPT(UG) o(1).
• Done !

48
Other Hardness Results

• Vertex Cover
• Friedguts Theorem
• Every boolean function with low average
  sensitivity is a junta.
• Sparsest Cut, Min-2SAT Deletion KahnKalaiLinial
  Every balanced boolean function has a
• 
  co-ordinate with influence log n/n.
• Bourgains Theorem (inspired by
  Hastad-Sudans 2-bit Long Code test)
• Every boolean function with low noise
  sensitivity is a junta.
• Coloring 3-Colorable MOO05 inspired.
• Graphs

49
Basic Paradigm by BGS95, Hastad97

• ? Hardness results for Clique, MAX-3SAT,
  .
• Instead of Unique Games, use reduction from
• general 2P1R Games (PCP Theorem Raz).
• Hypercube Bits in the Long Code Bellare
• 
  Goldreich Sudan95
• PCPs with 3 or more queries (testing Long
  Code).
• Not enough to construct 2-query PCPs.

50
Why UGC and not 2P1R Games?

• Power in simplicity.
• Obvious way of encoding a permutation
• constraint.
• Basic Gadget ----gt Bipartite Gadget with
• permutation.

51
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

52
I Hope UGC is True

• Implies all the right hardness
• results in a unifying way.
• Neat applications of Fourier theorems
• Bourgain02, KKL88, Friedgut98, MOO05
• Surprising application to theory of metric
• embeddings and SDP-relaxations KV05.
  
• Mere coincidence ?

53
Supporting Evidence

• Feige Reichman04
• Gap-Unique Game (C?, ?) is NP-hard.
• i.e. For every constant C, there is ?
  s.t.
• it is NP-hard to tell if a UG has
• OPT gt C ? or OPT lt ?.
• However C ? --gt 0 as ? --gt 0.

54
Supporting Evidence

• Khot Vishnoi05
• SDP relaxation for Unique Game
• has integrality gap (1-? , ?).

55
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

56
Disproving UGC means ..

• For small enough (constant) ?,
• given a UG with optimum 1- ?,
• algorithm that finds a labeling satisfying
• (say) 50 constraints.

57
Algorithmic Results

• Algorithm that finds a labeling
• satisfying f(?, k, n) fraction of
  constraints.
• Khot02 1- ?1/5 k2
• Trevisan05 1- ?1/3 log1/3 n
• Gupta Talwar05 1- ? log n
• CMM05 1/k? , 1- ?1/2 log1/2 k
  
• None of these disproves UGC.

58
Quadratic Integer Program For Unique Game Feige
Lovasz92
variable
u1 , u2 , , uk ? 0,1
u
? k ? k
v
k labels
v1 , v2 , , vk ? 0,1
vi 1 if Label(v) i 0 otherwise
59
Quadratic Program for Unique Games

• Constraints on edge-set E.
• Maximize ? ? ui vp(i)
• (u, v) ? E
  i1,2,..,k
• ? u ? i ? k, ui ? 0,1
• ? u ? ui2 1
• i
• ?u ? i ? j , ui uj 0

60
SDP Relaxation for Unique Games

• Maximize ? ? ?ui,
  vp(i) ?
• (u, v)
  ? E i1,2,..,k
• ? u ? i ? k, ui is a vector.
• ? u ? ui 2 ? 1
• i1,2,..,k
• ? u ? i?j ? k, ?ui, uj? 0

61
Feige Lovasz92

• OPT(G) ? SDP(G) ? 1.
• If OPT(G) lt 1, then SDP(G) lt 1.
• SDP(Gm) (SDP(G))m
• Parallel Repetition Theorem for UG
• OPT(G) lt 1 ? OPT(Gm) ? 0

62
Khot02 Rounding Algorithm
r
r
Random r
u
v

• Label(u) 2, Label(v) 2
• Pr Label(u) Label(v) gt 1 -
  ?1/5 k2
• Labeling satisfies 1 - ?1/5 k2 fraction
• of constraints in expected sense.

63
CMM05 Algorithm

• Labeling that satisfies 1/k? fraction
• of constraints. (Optimal KV05)

All i s.t. ui is close to r are
taken as candidate labels to u. Pick one
of them at random.
64
Trevisan05 Algorithm

• Given a unique game with optimum
• 1- 1/log n, algorithm finds a labeling
• that satisfies 50 of constraints.
• Limit on hardness factors achievable
• via UGC (e.g. loglog n for Sparsest Cut).
  

65
Trevisan05 Algorithm
Variables and constraints

• Leighton Rao88 Delete a few constraints
  and
• remaining graph has
  connected
• components of low
  diameter.

66
Trevisan05 Algorithm

• A good algorithm for graphs with low
• diameter.

67
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

68
Already Covered Lets move on .
69
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

70
KV05 Integrality Gaps for
SDP-relaxations

• MAX-CUT
• Sparsest Cut
• Unique Game
• Gaps hold for SDPs with Triangle Inequality.

71
Integer Program for MAX-CUT

• Given G(V,E)
• Maximize ¼ ? vi - vj 2
• (i,
  j) ? E
• ? i, vi ? -1,1
• Triangle Inequality (Optional) ? i,
  j , k,
• vi - vj 2 vj - vk 2 ?? vi
  - v k2

72
Goemans-Williamsons SDP Relaxation for MAX-CUT

• Maximize ¼ ? vi - vj
  2
• 
  (i, j) ? E
• ? i, vi ? Rn, vi 1
• Triangle Inequality (Optional) ? i, j
  , k,
• vi - vj 2 vj - vk 2 ??
  vi - v k2

73
Integrality Gap for MAX-CUT

• Goemans Williamson94
• Integrality gap ? 1/0.878..
• Karloff99 Feige Schetchman 01
• Integrality gap ? 1/0.878.. - ?
• SDP solution does not satisfy Triangle
  Inequality.
• Does Triangle Inequality make the SDP
  tighter ?
• NO if Unique Games Conj. is true !

74
Integrality Gap for Unique Games SDP
SDP(G) 1-o(1)
Unique Game G with OPT(G) o(1)
75
Integrality Gap for MAX-CUT with Triangle
Inequality
OPT(G) o(1)
PCP Reduction
No large cut
Good SDP solution
? u1 ? u2 ? u3 ? uk-1 ?
uk
-1,1k
76
Overview of the talk

• The UGC
• Hardness of Approximation Results
• I hope UGC is true
• Attempts to Disprove Algorithms
• Connections/applications
• Fourier Analysis
• Integrality Gaps
• Metric Embeddings

77
Metrics and Embeddings

• Metric is a distance function on n such that
• d(i, j) d(j, k) ? d(i, k).
• Metric d embeds into metric ? with
• distortion ? ? 1 if
• ? i, j d(i, j) ? ? (i, j) ? ? d(i, j).
  

78
Negative Type Metrics

• Given a set of vectors satisfying
  Triangle Inequality
• ? i, j , k,
• vi - vj 2 vj - vk 2 ??
  vi - v k2
• d(i, j) vi - vj 2 defines
  a metric.
• These are called negative type metrics.
  
• L1 ? NEG ? METRICS

79
NEG vs L1 Question

• Goemans, Linial 95
• Conjecture NEG metrics embed into L1
  
• with O(1)
  distortion.

O(1) Integrality Gap O(1) Approximation
Linial London Rabinovich94 Aumann
Rabani98
Chawla Krauthgamer Kumar Rabani Sivakumar
05 KV05 ?(1) hardness result
Unique Games Conjecture
Sparsest Cut
80
NEG vs L1 Lower Bound

• ?(loglog n) integrality gap for
  Sparsest
• Cut SDP. KhotVishnoi05,
  KrauthgamerRabani05
• ? A negative type metric that needs
• distortion ?(loglog n) to embed into
  L1.

81
Open Problems

• (Dis)Prove Unique Games Conjecture.
• Prove hardness results bypassing UGC.
• NEG vs L1 , Close the gap.
• ?(log log n) vs ?(?log n loglog n)
• Arora Lee
  Naor04

82
Open Problems

• Prove hardness of Min-Deletion version
• of Unique Games. (log n approx.
  GT05)
• Integrality gaps with k-gonal inequalities.
• Is hypercube (Long Code) necessary ?

83
Open Problems

• More hardness results, integrality gaps,
• embedding lower bounds, Fourier Analysis,
  

Samorodnitsky Trevisan05 Gowers
Uniformity, Influence of Variables, and
PCPs. UGC ? Boolean k-CSP is hard to
approximate within 2k- log k
Independent Set on
degree D graphs is hard to
approximate within D/poly(log
D).
84
Open Problems in Approximability

• Traveling Salesperson
• Steiner Tree
• Max Acyclic Subgraph, Feedback Arc Set
• Bin-packing (additive approximation)
• Recent progress on Edge Disjoint Paths
• Network
  Congestion
• Shortest
  Vector Problem
• Asymmetric
  k-center (log n)
• Group
  Steiner Tree (log2 n)
• Hypergraph
  Vertex Cover

85
Linear Unique Games

• System of linear equations mod k.
• x1 x3 2
• 3 x5 - x2 -1
• x2 5 x1 0

KKMO04 UGC ? UGC in the special case of
linear
equations mod k.
86
Variations of Conjecture

• 2-to-1 Conjecture K02
• ?-Conjecture DMR05
• ? NP-hard to color 3-colorable graphs
• with O(1) colors.

? ? k ? k
? ? k ? k